Gas turbine engines include a compressor section, a turbine section, and a combustor section. Many gas turbine engines also include a liftoff seal to prevent leakage (or flow) from one compartment to another compartment. The vibratory response of a liftoff seal may cause damage to the carbon seal. The fluid film formed between the carbon seal and the seal plate can produce negative damping under certain operating conditions, which can overcome the positive mechanical damping present in the system. When negative damping occurs, the carbon seal can impact the seal plate, causing carbon seal wear and damage. A typical approach for adding damping is to use a sheet metal damper on the bellows, which may not always damp the modes of interest that result in large carrier motion.